A model study of a dirigible (a large balloon that travels through the air) is to be performed. The 10-m-diameter dirigible is to travel at 20 m/s. If a 40-cm-diameter model is proposed for use in a wind tunnel, or a 10-cm-diameter model for use in a 20°C water channel, which should be selected?

Question

A model study of a dirigible (a large balloon that travels through the air) is to be performed. The 10-m-diameter dirigible is to travel at 20 m/s. If a 40-cm-diameter model is proposed for use in a wind tunnel, or a 10-cm-diameter model for use in a 20°C water channel, which should be selected? Suppose that the wind-tunnel model is used with a speed of 15 m/s and a drag force of 3.2 N is measured. What force would be expected on the water-channel model with a speed of 2.4 m/s in the water channel? What horsepower would be predicted to overcome drag on the prototype? Assume that the flow is Reynolds number independent for Re > [latex]10^5[/latex].

Accepted Answer

[latex]Re_p =\frac{20 imes 10}{1.5 imes 10^{-5}}=13.3 imes 10^6 [/latex]. This is a high-Reynolds-number flow.

For the wind tunnel, let [latex]Re_m =10^5 = V_m ×0.4/1.5×10^{−5}. ∴V_m ≥ 3.75 m/s[/latex]

For the water channel, let [latex]Re_m =10^5 = V_m ×0.1/1×10^{−6}. ∴V_m ≥1.0 m/s[/latex]

[latex]\underline{Either could be selected}[/latex] . The more convenient facility would be chosen.

[latex]\frac{F_{m1}}{F_{m2}} = \frac{ ho _{m1}V ^2_{m1}\ell _{m1}^2 }{ ho _{m2}V ^2_{m2}\ell _{m2}^2}. ∴ F_{m2}=3.2 \frac{1000}{1.23} \frac{2.4^2}{15^2} imes \frac{0.1^2}{0.4^2} =\underline{4.16 N}[/latex] .

[latex]\frac{\dot{W}_m }{\dot{W}_p } = \frac{ ho _mV ^3_m\ell _m^2 }{ ho _pV ^3_p\ell ^2_p} =\frac{15^3 imes 0.4^2}{20^3 imes 10^2} . ∴\dot{W}_p =(15 imes 3.2) \frac{20^3}{15^3} imes \frac{10^2}{0.4^2}=71100 W or \underline{ 95 hp } [/latex].

Question 6.62: A model study of a dirigible (a large balloon that travels t...

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